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उपसूत्र १. आनुरूप्येण

(Upasūtra 1. ānurūpyeṇa) - Proportionately.

The Upasūtra: ānurūpyeṇa (Proportionately) is used for multiplication (x × y) of two numbers, where both the numbers are near a common base of multiples of power of 10 (m10n). This Upasūtra is similar to multiplication using Sūtra 2. nikhilaṃ navataścaramaṃ daśataḥ (discussed here »).
 
Note, that this Upasūtra is applicable in a similar condition of the multiplication of number with Sūtra 2. nikhilaṃ navataścaramaṃ daśataḥ, where both the numbers are near a common base of power of 10 (10n), and not multiples of power of 10 (m10n).
For example, if we are to multiply: 469 × 487 with Sūtra 2. nikhilaṃ navataścaramaṃ daśataḥ, considering a base of 1000 - we will be required to multiply: (1000 - 469) × (1000 - 487) = 531 × 513, which is no better than the initial problem. This is where this Upasūtra: ānurūpyeṇa, comes to rescue.
 
As an illustration, let us use this Upasūtra for:
42 × 47
  Steps 42 × 47
1. Find a Base, that is near both the Operands - that is a multiple of power of 10. In this case, Base = 50 = 5 × 10
2. Write the differences from the Base on the right hand side of the respective numbers. In this case, the differences are 42 = 50 - 8, and 47 = 50 - 3
 
×  42   -8
×  47   -3
-----------
3. Derive the result in two parts. The RHS being the product of the diffences, and the LHS being (Any of the Original Numbers) added with the other number's difference. In this case,
×  42   -8
×  47   -3
-----------
× [39 , 24]
 
Note: 8 × 3 = 24, and 42 - 3 = 39
4. Convert the number to Base10, using the same proportion as Base10 is to the considered Base. In this case, Base10:Base50 = 1:5
[39,24] in Base50 =
[39 × 5, 24] in Base10
= [195,24]
= 1950 + 24
= 1974, which is the answer!
Now, let us take another example:
43 × 52
  Assuming Base50,
×  43   -7
×  52   +2
-----------
× [45 ,-14]
 
Note: -7 × 2 = -14, and 43 + 2 = 45
 
[45,-14] in Base10 = [45 × 5,-14]
= [225,-14]
= 2250 - 14
= 2236, which is the answer!
So, for a practitioner of Vedic Mathematics, for something like:
469 × 487
  Assuming Base500,
×  469   -31
×  487   -13
-------------
× [456  ,403]
 
Note: -31 × -13 = 403, and 469 - 13 = 456
 
[456,403] in Base10 = [456 × 50,403]
= [22800,403]
= 228000 + 403
= 228403
 
Thus, 469 × 487 = 228,403
 
Again, for something like:
2988 × 3014
  Assuming Base3000,
×  2988   -12
×  3014   +14
--------------
× [3002 ,-168]
 
Note: -12 × 14 = -168, and 2988 + 14 = 3002
 
[3002,-168] in Base10
= (3002 × 3000) - 168
= 9006000 - 168 = 9005832
 
Thus, 2988 × 3014 = 9,005,832
Obviously, the closer the numbers are to the base, the easier it becomes. Taking it further, for something like:
49999 × 50003
  Assuming Base50000,
×  49999   -1
×  50003   +3
--------------
× [50002  ,-3]
 
Note: -1 × 3 = -3, and 50003 - 1 = 50002
 
[50002,-3] in Base10
= (50002 × 50000) - 3
= 250010,0000 - 3 = 250009,9997
 
Thus, 49999 × 50003 = 2,500,099,997
But, why does it work? For this Upasūtra (ānurūpyeṇa), let us consider the following:
  Assuming two numbers, x and y
Assuming any Base, x = Base + a, and y = Base + b
Then, x × y = (Base + a)(Base + b)
= (Base)2 + a × Base + b × Base + ab
= (Base)2 + Base (a + b) + ab
 
But, Base = x - a = y - b
So, substituting Base = (x - a)
= Base (x - a) + Base (a + b) +ab
= Base (x - a + a + b) + ab
= Base (x - b) + ab
 
This is exactly what this Upasūtra makes us do.
 
Note that, if the Base is 10 - no further conversion is required. For any other Base, it is converted to Base10.
 
Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »
 
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